Data communication in systems where channel conditions include time varying reflections and/or scattering of the transmitted signal wave is generally more difficult than in systems where a time-invariant signal path exists between the transmitter and the receiver. Fading results when multiple paths from random reflections and/or scattering combine to cancel much of the transmitted signal. A fading channel is said to be dispersive in frequency if the fading is not constant over the signal band of interest. A frequency-dispersive channel when excited by an impulse function in time produces multipath delayed received signals spread out in the delay dimension. The average power of these multipath delayed signals versus delay represents the multipath profile of the frequency dispersive channel. The twice-standard deviation 2σD in units of time is a statistical measure of the delay spread of the fading channel. A fading channel is said to be dispersive in time if the fading changes with time. A time-dispersive channel when excited by a sinusoid produces multiple received signals spread out in the frequency dimension. Analogous to the delay spread, the twice-standard deviation 2σf in Hz of the average power vs. frequency profile is a measure of the frequency spread.
Because of the fading, weak signal-to-noise ratio conditions will occur at the receiver so communication techniques in this application are generally restricted to modulation choices with a small number of bits per modulation symbol. Practical systems use Quadrature Phase-Shift-Keying (QPSK) with 2 bits per symbol and 8 Phase-Shift-Keying (8PSK) with 3 bits per symbol. High data rates then can only be realized with corresponding high modulation symbol rates. This results in a symbol period T that can be less than the delay spread 2σD and a symbol rate 1/T that is much greater than the frequency spread 2σf. The frequency selective fading when 2σD/T is near or exceeds unity results in intersymbol interference (ISI) and a potential for additional diversity. The high symbol rate with 1/T>>2σf insures that there are many symbols within an interval where the fading is not changing. Under this condition adaptation by estimating either the channel or receiver processor parameters is possible.
The weak signal-to-noise ratio conditions also often require additional redundant, i.e., diversity, paths. For example, space diversity is achieved with redundant paths provided by extra antennas and frequency diversity is achieved with redundant paths provided by additional signal bands with the same transmitted signal information. When the diversity is realized by multipath effects associated with the frequency dispersive fading, it is referred to as implicit diversity because it is inherent in the channel. In contrast explicit diversity systems such as space or frequency are at the communication designer's discretion.
Examples of fading dispersive channels include tropospheric scatter (troposcatter) systems, high frequency (HF) ionospheric systems, and cellular radio systems where the reflections and scattering are produced by buildings, trees, and other physical objects. Nominal spread values for these example systems are
Delay SpreadFrequency Spread2σD (1E-6 sec.)2σf (Hz)Troposcatter0.21.0HF1000.00.1Cellular4.010.0
One observes, for example, a high data rate system with T<2σD and 1/T>>2σf in an HF system would not be a high data rate system in a troposcatter or cellular system.
In these radio system fading channel applications, the transmitted high data rate is generally required to be contained within a certain frequency band allocation. Limitation of spectral emissions is generally accomplished with a spectrum control transmitter filter. With present technology the filter can be realized as a Finite impulse response (FIR) filter with multiple coefficients per symbol and a filter span of multiple symbols. The spectrum control filter can be designed so that there is no ISI when the fading channel is not frequency dispersive. However under general fading dispersive conditions ISI is produced that is due to the combination of the spectrum control filter and the frequency dispersive channel.
In Feedback Equalization for Fading Dispersive Channels, P. Monsen, IEEE Trans. On Information Theory, pp. 56-64, January 1971, (hereafter Feedback Equalization) the optimum infinite length Decision-Feedback Equalizer (DFE) is developed under a Minimum Mean-Square-Error (MMSE) criterion. The length corresponds to the number of taps on a tapped-delay-line (TDL) filter. Additionally a finite length DFE is presented and an adaptation method based on a stochastic gradient technique is used to track the time-dispersive effects. The infinite length DFE contains a matched filter and forward TDL filter combination that processes the received signal and a backward TDL filter that processes receiver decisions. The forward and backward TDL filters have tap spacing equal to the symbol period T. The optimum solution requires that the forward TDL filter be of infinite length. In a suboptimum but finite solution in Feedback Equalization the matched filter/forward TDL filter combination is realized with a single finite length TDL filter with tap spacing equal to the Nyquist interval 1/B, where for practical systems 1/B<T. This truncated finite length DFE is suboptimum because the number of intersymbol interferers is equal to or greater than the symbol dimensions of the 1/B TDL filter so that all interference can only be cancelled in the infinite length limit. Notwithstanding this limitation, the finite length DFE in Feedback Equalization was shown to be able to cope with the ISI produced by the delay spread while extracting implicit diversity with the important result that the net effect of the frequency selective fading was to improve performance rather than degrade it.
In Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference, G. D. Forney Jr., IEEE Trans. On Information Theory, pp. 363-378, May 1972 the Maximum Likelihood Sequence Estimator (MLSE) was developed for the general ISI problem. The MLSE is more complex but superior to the DFE because the MLSE optimization is based on bit error rate probability rather than MMSE. This superiority is most important in channels with nearly perfect ISI cancellation. However it has been shown in studies, for example Adaptive equalization of the slow fading channel, P. Monsen, IEEE Trans. Communications, August 1974, that these types of perfect ISI canceling channels do not occur that often in fading channel applications and there is little or no advantage of using the more complex MLSE technique.
The suboptimum finite length DFE of Feedback Equalization was developed in a military troposcatter application, Megabit digital troposcatter subsystem, C. J. Grzenda, D. R. Kern, and P. Monsen, Proc. Natl Telecommunication Conf., pp. 28-15 to 28-19, New Orleans La., December 1975. This DFE modem operated in a quadruple diversity configuration with for each diversity a forward TDL filter with three taps at a tap spacing of T/2. The backward TDL filter employed five taps at spacing T. The highest data rate of 12.6 Mb/s was contained in a 99% bandwidth of 15 Mhz, i.e. 0.84 bits/sec/Hz. The modem was tested with a channel simulator with values of 2σD/T that exceeded unity. The modem complexity in this application was dominated by the diversity order and the associated complexity of the forward TDL filter for each diversity. The use of the stochastic gradient algorithm also required continuous transmission so that this DFE could not be used in frame transmission systems with abrupt changes such as time-division multiplexed packet systems or frequency-hopping applications.
Block equalization strategies can be used in fading dispersive channels when the frequency spread 2σf is much less than the reciprocal block length such that the channel can be viewed as stationary during the block interval. A block DFE is described in Channel Equalization for Block Transmission Systems, G. K. Kaleh, IEEE Journal Selected Areas in Comm., pp. 110-121, January 1995, Known symbols are imbedded in the block in order to act as a time guard between successive blocks and to facilitate channel identification. The DFE is composed of a threshold detector, a feedforward transformation, and a feedback transformation. The DFE follows a symbol rate sampler and- a filter matched to -the combined transmit and unknown channel filter. The article does not include a channel identification technique. Another block DFE is described in Block Channel Equalization in the Presence of a Cochannel Interferent Signal, A. Ginesi, G. M. Vitetta, and D. D. Falconer, IEEE Journal Selected Areas in Comm., pp. 1853-1862, November 1999. In this solution an ideal anti-alias receive filter and a feedforward transformation with half-symbol tap spacing is used to reduce the effects of both ISI and cochannel interference. The feedforward transformation in the DFE is based on the suboptimum truncated infinite length DFE of Feedback Equalization and additionally the functions of matched filtering and ISI reduction are not separated to exploit transmitter fixed filter characteristics. In this and other block equalization techniques the equalizer complexity is a function of the block length and not the channel delay spread. In high data rate systems the block length is generally much greater than the delay spread thus producing an inherent block detection disadvantage.
In MMSE Decision-Feedback Equalizers: Finite-Length Results, N. Al-Dhahir and J. M. Cioffi, IEEE Trans. Information Theory, July, 1995 (hereafter, Finite Length DFE) the optimum finite length DFE under known channel conditions is constructed from a vector TDL filter. A forward vector TDL filter is determined by a Cholesky factorization of the matrix cross correlation of the received samples and transmitted symbols. In Fast computation of channel-estimate based equalizers in packet data transmission, N. Al-Dhahir and J. M. Cioffi, IEEE Trans. on Signal Processing, pp. 2462-2473, November 1995 (hereafter Fast Computation) an efficient algorithm is presented for computing the parameters of the finite length DFE assuming that the channel has been estimated using a known training pattern. Both these papers are for a single channel packet system and do not consider efficient implementation for multiple channels in diversity applications. A spectrum control filter in the transmitter and a corresponding noise-limiting filter in the receiver and their collective effect on channel estimation and equalization are also not considered in Finite Length DFE and Fast Computation.
Frequency and space diversity systems in, for example, troposcatter systems require separate redundant paths for each diversity reception. For example a quadruple space diversity system would use two transmit antennas transmitting on antenna ports with two orthogonal polarizations and two receive antennas with four radio frequency down-converters for each of the four antenna/polarization combinations. The polarization orthogonality does not in itself provide uncorrelated paths through the fading dispersive channel but quadruple diversity is still achieved as a result of the four separate paths between the transmit and receive antenna pairs. In this example the polarization orthogonality allows separation of these four paths at the receiver. A transmit diversity technique that separates transmissions from different antennas using a space-time block code (STBC) has been suggested in A Simple Transmit Diversity Technique for Wireless Communications, S. M. Alamouti, IEEE Journal Selected Areas in Comm., pp. 1451-1458, October 1998 (hereafter Transmit Diversity). However it was shown in Performance Evaluation and Analysis of Space-Time Coding in Unequalized Multipath Fading Links, Y. Gong and K. B. Letaief, IEEE Trans. Communications, pp. 1778-1782, November 2000 that an error floor occurs if nondispersive standard receivers are used when STBC signals are corrupted by ISI. In a direct approach, equalization to correct for the ISI degradation is separated from the STBC decoding. Examples of this direct approach include Multiple Input/Multiple Output (MIMO) equalization for space-time block coding, Proc. IEEE Pacific Rim Conf. Communications, Computers, Signal Processing, 1999, pp.341-344 and Finite-length MIMO Decision-Feedback Equalization for Space-Time Block-coded Signals over Multipath Fading Channels, N. Al-Dhahir, et.al., IEEE Trans. Vehicular Tech., pp. 1176-1182, July 2001. In this direct solution for the most relevant STBC application of a single receive antenna, the zero-forcing solution where all ISI is cancelled does not exist and the MMSE DFE has poor performance. An integrated equalization/STBC decoding technique using a widely linear (WL) processor is described in Equalization Concepts for Alamouti's Space-Time Block Code, W. H. Gerstacker, et. al., IEEE Trans. Communications, pp. 1178-1190, July 2004. The WL processor operates on both the received signal and its complex conjugate so the effects of complex conjugation in the STBC codebook can be considered. Adaptation of the processor and equalization with sampling greater than the symbol rate in order to meet the Nyquist criterion are not considered in this article. Another integrated equalization/STBC solution that includes a method of training-aided channel estimation is given in Transmit Diversity and Linear and Decision-Feedback Equalizations for Frequency-Selective Fading Channels, Ling Li, et. al., IEEE Trans. Vehicular Tech., pp. 1217-1231, September 2003. The DFE in this article uses a linear equalizer estimate at the end of a subframe to begin decision cancellation for earlier estimates in the subframe. Also the channel estimation and equalization in the Li article do not separate the transmit and channel filter effects and the article does not provide an equalization solution for required Nyquist sampling at greater than the symbol rate.
Although the techniques described above have been used for improving the quality of high data rate communication in fading dispersive channels, it has been recognized that MLSE technigues are more complex with little additional improvement relative to the DFE, block equalization techniques can be more complex in high data rate systems, and previous DFE techniques do not provide an optimal finite length solution under conditions of spectral emission limits, unknown channel variations, and diversity transmissions. Also systems operating in fading dispersive channels usually have performance significantly poorer than systems, such as satellite systems, that operate over nonfading channels. Additionally equalization is difficult in fading dispersive channel systems using transmit diversity transmissions with space-time block coding.